TimeDelayARACNE: Reverse engineering of gene networks from timecourse data by an information theoretic approach
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Abstract
Background
One of main aims of Molecular Biology is the gain of knowledge about how molecular components interact each other and to understand gene function regulations. Using microarray technology, it is possible to extract measurements of thousands of genes into a single analysis step having a picture of the cell gene expression. Several methods have been developed to infer gene networks from steadystate data, much less literature is produced about timecourse data, so the development of algorithms to infer gene networks from timeseries measurements is a current challenge into bioinformatics research area. In order to detect dependencies between genes at different time delays, we propose an approach to infer gene regulatory networks from timeseries measurements starting from a well known algorithm based on information theory.
Results
In this paper we show how the ARACNE (Algorithm for the Reconstruction of Accurate Cellular Networks) algorithm can be used for gene regulatory network inference in the case of timecourse expression profiles. The resulting method is called TimeDelayARACNE. It just tries to extract dependencies between two genes at different time delays, providing a measure of these dependencies in terms of mutual information. The basic idea of the proposed algorithm is to detect timedelayed dependencies between the expression profiles by assuming as underlying probabilistic model a stationary Markov Random Field. Less informative dependencies are filtered out using an auto calculated threshold, retaining most reliable connections. TimeDelayARACNE can infer small local networks of time regulated genegene interactions detecting their versus and also discovering cyclic interactions also when only a mediumsmall number of measurements are available. We test the algorithm both on synthetic networks and on microarray expression profiles. Microarray measurements concern S. cerevisiae cell cycle, E. coli SOS pathways and a recently developed network for in vivo assessment of reverse engineering algorithms. Our results are compared with ARACNE itself and with the ones of two previously published algorithms: Dynamic Bayesian Networks and systems of ODEs, showing that TimeDelayARACNE has good accuracy, recall and Fscore for the network reconstruction task.
Conclusions
Here we report the adaptation of the ARACNE algorithm to infer gene regulatory networks from timecourse data, so that, the resulting network is represented as a directed graph. The proposed algorithm is expected to be useful in reconstruction of small biological directed networks from time course data.
Keywords
Mutual Information Positive Predictive Value Bayesian Network Dynamic Bayesian Network Yeast Cell CycleBackground
In order to understand cellular complexity much attention is placed on large dynamic networks of coregulated genes at the base of phenotype differences. One of the aims in molecular biology is to make sense of highthroughput data like that from microarray of gene expression experiments. Many important biological processes (e.g., cellular differentiation during development, aging, disease aetiology etc.) are very unlikely controlled by a single gene instead by the underlying complex regulatory interactions between thousands of genes within a fourdimension space. In order to identify these interactions, expression data over time can be exploited. An important open question is related to the development of efficient methods to infer the underlying gene regulation networks (GRN) from temporal gene expression profiles. Inferring, or reverseengineering, gene networks can be defined as the process of identifying gene interactions from experimental data through computational analysis. A GRN can be modelled as a graph G = (V, U, D), where V is the set of nodes corresponding to genes, U is the set of unordered pair (undirected edges) and D is the set of ordered pairs D (directed edges). A directed edge d_{ ij }from v_{ i }to v_{ j }is present iff there is a causal effect from node v_{ i }to node v_{ j }. An undirected edge u_{ ij }represents the mutual association between nodes v_{ i }and v_{ j }. Gene expression data from microarrays are typically used for this purpose. There are two broad classes of reverseengineering algorithms [1]: those based on the physical interaction approach which aim at identifying interactions among transcription factors and their target genes (genetosequence interaction) and those based on the influence interaction approach that try to relate the expression of a gene to the expression of the other genes in the cell (genetogene interaction), rather than relating it to sequence motifs found in the promoters. We will refer to the ensemble of these influence interactions as gene networks. Many algorithms have been proposed in the literature to model gene regulatory networks [2] and solve the network inference problem [3].
Ordinary Differential Equations
Reverseengineering algorithms based on ordinary differential equations (ODEs) relate changes in gene transcript concentration to each other and to an external perturbation.
Typical perturbations can be for example the treatment with a chemical compound (i.e. a drug), or the over expression or down regulation of particular genes. A set of ODEs, one for each gene, describes gene regulation as a function of other genes. As ODEs are deterministic, the interactions among genes represent causal interactions, rather than statistical dependencies. The ODEbased approaches yield signed directed graphs and can be applied to both steadystate and timeseries expression profiles [3, 4].
Bayesian Networks
A Bayesian network [5] is a graphical model for representing probabilistic relationships among a set of random variables X_{ i }, where i = 1, ⋯, n. These relationships are encoded in the structure of a directed acyclic graph G, whose vertexes (or nodes) are the random variables X_{ i }. The relationships between the variables are described by a joint probability distribution P(X_{1}, ⋯, X_{ n }). The genes, on which the probability is conditioned, are called the parents of gene i and represent its regulators, and the joint probability density is expressed as a product of conditional probabilities. Bayesian networks cannot contain cycles (i.e. no feedback loops). This restriction is the principal limitation of the Bayesian network model [6]. Dynamic Bayesian networks overcome this limitation [7]. Dynamic Bayesian networks are an extension of Bayesian networks able to infer interactions from a data set consisting of timeseries rather than steadystate data.
Graphical Gaussian Model
Graphical Gaussian model, also known as covariance selection or concentration graph models, assumes multivariate normal distribution for underlying data. The independence graph is defined by a set of pairwise conditional independence relationships calculated using partial correlations as a measure of independence of any two genes that determine the edgeset of the graph [8]. Partial cross correlation has been also used to deal with time delays [9].
Gene Relevance Network
Gene relevance networks are based on the covariance graph model. Given a measure of association and defined a threshold value, for all pairs of domain variables (X, Y), association A(X, Y) is computed. Variables X and Y are connected by an undirected edge when association A(X, Y) exceeds the predefined threshold value. One of the measures of association is the mutual information (MI) [10, 11], one of the information theory (IT) main tools. In IT approaches, the expression level of a gene is considered as a random variable. MI is the main tool for measuring if and how two genes influence each other. MI between two variables X and Y is also defined as the reduction in uncertainty about a variable X after observing a second random variable Y. Edges in networks derived by informationtheoretic approaches represent statistical dependencies among gene expression profiles. As in the case of Bayesian network, the edge does not represent a direct causal interaction between two genes, but only a statistical dependency. It is possible to derive the informationtheoretic approach as a method to approximate the joint probability density function of gene expression profiles, as it is performed for Bayesian networks [12, 13, 14].
TimeCourse Reverse Engineering
Summary of the Proposed Algorithm
TimeDelayARACNE tries to extend to timecourse data ARACNE (Algorithm for the Reconstruction of Accurate Cellular Networks) retrieving time statistical dependency between gene expression profiles. The idea on which TimeDelayARACNE is based comes from the consideration that the expression of a gene at a certain time could depend by the expression level of another gene at previous time point or at very few time points before. TimeDelayARACNE is a threesteps algorithm: first it detects, for all genes, the time point of the initial changes in the expression, secondly there is network construction and finally a network pruning step. Is is worth noticing that, the analytical tools for time series often require conditions such as stability and stationarity (see [24]). Although it is not possible to state that these conditions hold in general for microarray data, this is due to the limited number of samples and to the particular experimental setup producing the data, nevertheless time series analysis methods have been demonstrated to be useful tools in many applications of time course gene expression data analysis, for example Ramoni et al. [25], used an autoregressive estimation step as feature extraction prior to classification, while Holter et al., [26] use the characteristic modes obtained by singular value decomposition to model a linear framework resulting in a time translational matrix. In particular TimeDelayARACNE, just as many related works (see for example the paper of [27]) implicitly assumes stationarity and stability conditions in the kernel regression estimator used for the computation of the mutual information, as described in the section Methods. Indeed, the synthetic data generation model (4) and (5) assumes a weakly stationary linear autoregressive time series. We do not attempt removal of the trend because of the short length of the data and the wide variability of the periodicity of the cell division cycle.
Results and Discussion
Algorithm Evaluation
TimeDelayARACNE was evaluated first alone than against ARACNE, dynamical Bayesian Networks implemented in the Banjo package [28] (a software application and framework for structure learning of static and dynamic Bayesian networks) and ODE implemented in the TSNI package [29] (Time Series Network Identification) with both simulated gene expression data and real gene expression data related to yeast cell cycle [30], SOS signaling pathway in E. coli[31] and an in vivo synthetic network, called IRMA [32]. Details on the gene expression data and the construction of the simulated networks are presented in Methods section.
Synthetic Data
In order to quantitatively evaluate the performance of the algorithm reported here over a dataset with a simple and fair "gold standard" and also to evaluate how the performance depend of the size of the problem at the hand, such as network dimension, number of time points, and other variables we generated different synthetic datasets. Our profile generation algorithm (see Methods) starts by creating a random graph which represents the statistical dependencies between expression profiles, and then the expression values are generated according to a set of stochastic difference equation with random coefficients. The network generation algorithm works in such a way that each node can have zero (a "stimulator" node) one or two regulators. In addition to the random coefficients of the stochastic equations, a random Gaussian noise is added to each expression value. The performance are evaluated for each network size, number of time points and amount of noise by averaging the PPV, recall and Fscore over a set of 20 runs with different randomly generated networks. The performance is measured in terms of:
 positive predictive value (PPV), it is the percentage of inferred connections which are correct:
 recall, it is the percentage of true connection which are correctly inferred by the algorithm:
 Fscore. Indeed, the overall performance depend both of the positive prediction value and recall as one can improve the former by increasing the detection threshold, but at the expenses of the latter and vice versa. The Fscore measure is the geometric mean of p and r and represents the compromise between them:
Since TimeDelayARACNE always tries to infer edge's direction, so the precisionrecall curves take into account direction. As a matter of fact an edge is considered as a true positive only if the edge exist in reference table and the direction is correct.
TimeDelayARACNE, TSNI, Banjo and ARACNE performance against synthetic data changing network gene numbers.
TimeDelayARACNE  TSNI  Banjo  ARACNE  

Genes  Time Points  PPV  Recall  F score  PPV  Recall  F score  PPV  Recall  F score  PPV  Recall  F score 
genes 10  points 50  0.33  0.60  0.41  0.43  0.11  0.18  <0.19  0.13  0.15  0.78  0.12  0.21 
genes 20  points 50  0.46  0.35  0.39  0.52  <0.1  0.11  <0.1  0.13  0.11  0.55  0.06  0.10 
genes 30  points 50  0.47  0.23  0.29  0.35  <0.1  <0.1  <0.1  <0.1  <0.1  0.68  0.04  0.08 
TimeDelayARACNE, TSNI, Banjo and ARACNE performance against synthetic data changing network data points.
TimeDelayARACNE  TSNI  Banjo  ARACNE  

Genes  Time Points  PPV  Recall  F score  PPV  Recall  F score  PPV  Recall  F score  PPV  Recall  F score 
genes 10  points 10  0.24  0.22  0.22  0.29  0.14  0.19  <0.1  <0.1  <0.1  0.27  0.14  0.17 
genes 10  points 20  0.31  0.34  0.32  0.22  0.10  0.14  0.20  <0.1  0.13  0.25  0.09  0.13 
genes 10  points 30  0.31  0.42  0.34  0.39  0.11  0.17  0.17  0.10  0.12  0.63  0.14  0.22 
genes 10  points 40  0.41  0.55  0.45  0.39  <0.1  0.15  0.22  0.16  0.18  0.79  0.14  0.23 
genes 10  points 50  0.33  0.60  0.41  0.43  0.11  0.18  0.19  0.13  0.15  0.78  0.12  0.21 
Real Expression Profiles
In order to test TimeDelayARACNE performance on expression profiles we selected an eleven genes network from yeast S. cerevisiae cell cycle, more precisely part of the G1 step. Selected genes are: Cln 3, Cdc 28, Mbp 1, Swi 4, Clb 6, Cdc 6, Sic 1, Swi 6, Cln 1, Cln 2, Clb 5. To try to infer the gene network controlling yeast cell cycle regulation, we choose genes whose mRNA levels respond to the induction of Cln3 and Clb2 that are two wellcharacterized cell cycle regulators [33]. Late in G1 phase, the Cln3Cdc28 protein kinase complex activates two transcription factors, MBF (Mbp1 and Swi6) and SBF (Swi4 and Swi6), and these promote the transcription of some genes important for budding and DNA synthesis [34]. Entry into S phase requires the activation of the protein kinase Cdc28p through binding with cyclin Clb5 or Clb6, as well as the destruction of the cyclindependent kinase inhibitor Sic1 [35]. Swi4 associates with Swi6 to form the SCBbinding factor complex that activates G1 cyclin genes CLN1 and CLN2 in late G1. Mbp1, a transcription factor related to Swi4, forms the MCBbinding factor complex with Swi6, which activates DNA synthesis genes and Sphase cyclin genes CLB5 and CLB6 in late G1 [36]. In budding yeast, commitment to DNA replication during the normal cell cycle requires degradation of the cyclindependent kinase (CDK) inhibitor Sic1. The G1 cyclinCDK complexes Cln1Cdk1 and Cln2Cdk1 initiate the process of Sic1 removal by directly catalyzing Sic1 phosphorylation at multiple sites [37]. In Figure 3 we report network graphs reconstructed by the TimeDelayARACNE, TSNI and Banjo. We also report the KEGG pathway of the cellcycle in yeast. We consider this last information as a true table to compare the results of the algorithm with respect to the others. TSNI and Banjo are used with default settings reported in their manuals. TimeDelayARACNE recovers many genegene edges as reported in Table 3. We don't use PPV, recall and Fscore to evaluate the algorithm. Differences between true table and inferred network could be eventually due to the possible incongruence between experimental data and true table. We also tested the proposed algorithm using eight genes by E. coli SOS pathway [31]. In the E. coli after the cell is exposed to DNA damaging agents there is the activation of the SOS pathway. Such DNA damaging involves the induction of about 30 genes [38]. Many of these gene products are involved in DNA damage tolerance and repair (e.g. recA, lexA, umuDC, polB, sulA, and uvrA). The SOS response to DNA damage requires the recA and lexA gene products. Near the promoters of the SOS response genes there is a site (the SOS box) bonded by the repressor protein LexA that interferes with the binding of RNA polymerase [39]. Selected genes are: uvrD, lexA, umuDC, recA, uvrA, uvrY, ruvA, polB as in [40]. In Figure 4 we report the SOS pathway reconstruction by the three algorithms and the relative bibliographic control. In Table 4 there is a detailed description of the eight genes network connections showing that TimeDelayARACNE recovers these network topologies better than other algorithms.
TimeDelayARACNE test on the yeast eleven genes network.
Genes  Kegg  Correct  Wrong 

Cln3  Swi4, Swi6, Mbp1  Swi4, Swi6, Mbp1   
Swi6  Cln1/2, Clb5/6  Cln1, Clb6  Clb5 
Swi4  Cln1/2, CDC28  Cln1  Clb6 
Mbp1  Clb5/6, Cdc28     
Cln1  Sic1  Sic1   
Cln2  Sic1, Swi4/6  Sic1, Swi4   
Clb5  Cdc6    Sic1 
Clb6  Cdc6  Cdc6   
Sic1  Clb5/6, Cdc28     
Cdc6      Sic1 
Cdc28  Swi4/6, Mbp1, Cln1/2, Sic1, Cdc6  Swi6, Mbp1  Clb5 
TimeDelayARACNE test on the E. coli eight genes network.
Genes  SOS True Relations  Correct  Wrong 

recA  lexA     
lexA  uvrD, umuDC, recA, uvrA, uvrY, ruvA, polB.  uvrD, umuDC, recA, uvrA, ruvA, polB.   
uvrY      lexA 
TimeDelayARACNE test on the IRMA in vivo synthetic network.
TimeDelayARACNE  TSNI  Banjo  ARACNE  

PPV  Recall  F score  PPV  Recall  F score  PPV  Recall  F score  PPV  Recall  F score  
Switch ON network  
5 genes network  0.71  0.67  0.69  0.80  0.50  0.61  0.30  0.25  0.27  0.5  0.6  0.54 
simplified network  0.80  0.67  0.73  1.0  0.67  0.80        0.5  0.5  0.5 
Switch OFF network  
5 genes network  0.37  0.60  0.46  0.60  0.38  0.46  0.6  0.38  0.46  0.25  0.33  0.28 
simplified network  0.50  0.75  0.60  0.75  0.5  0.60        0.5  0.6  0.54 
According to this we try a compromise: no threshold was applied but we just apply the DPI pruning. We can observe that TimeDelayARACNE reaches good performance, in terms of recall and Fscore, in all considered cases with respect to the results reported in [32] and to the ARACNE results over the same networks. This means that it reaches a good compromise between PPV and recall.
Conclusions
The goal of TimeDelayARACNE is to recover gene time dependencies from timecourse data producing oriented graph. To do this we introduce time Mutual Information and Influence concepts. First tests on synthetic networks and on yeast cell cycle, SOS pathway data and IRMA give good results but many other tests should be made. Particular attention is to be made to the data normalization step because the lack of a rule. According to the little performance loss linked to the increasing gene numbers shown in this paper, next developmental step will be the extension from littlemedium networks to medium networks.
Methods
Datasets
Simulated Gene Expression Data
We construct some synthetic gene networks in order to compute the functions p, r and Fscore of the method having reference true tables and to compare its performance to other methods. According to the terminology in [41] we consider a gene network to be welldefined if its interactions allow to distinguish between regulator genes and regulated genes, where the first affect the behaviour of the second ones. Given a well defined network, we can have genes with zero regulators (called stimulators, which could represent the external environmental conditions), genes with one regulator, genes with two regulators, and so on. If a gene has at least one regulator (it is not a stimulator) then it owns a regulation function which describes its response to a particular stimulus of its regulator/regulators.
Our synthetic networks contain some stimulator genes with a random behaviour and regulated genes which can eventually be regulators of other genes. The network dynamics are modeled by linear causal relations which are formulated by a set of randomly generated equations. In particular, let us call the expression of gene i at time t as Open image in new window , our synthetic network generation module works as follows,

if gene i is a stimulator gene then its expression profile, Open image in new window , t = 0, 1, ... is randomly initialized with a sequence of uniform numbers in [1, 100].

for each nonstimulator gene i, Open image in new window is initialized with a uniform random number in [1,100]
 for each nonstimulator gene i, the expression values Open image in new window , ..., Open image in new window are computed according to a stochastic difference equation with random coefficients depending on one or two other regulator genes by using one of the two equations below:
here the coefficients Open image in new window and Open image in new window are random variables uniformly distributed in [0, 1] and Open image in new window is a random Gaussian noise with zero mean and variance σ^{2}. Moreover the regulators genes p_{ i }and q_{ i }of the ith are randomly selected at the beginning of each simulation run. The network generation algorithm is set in such a way that 75% of genes have one regulator and 25% of genes have two regulators.

each expression profile is then normalized to be within the interval [0, 1]
In our experiments, the PPV, recall and Fscore of the proposed and the other methods is computed as the average over a set of 20 runs over different random networks with the same number of genes, number of time points and noise levels.
Microarray Expression Profiles
The time course profiles for a set of 11 genes, part of the G1 step of yeast cell cycle, are selected from the widely used yeast, Saccharomyces cerevisiae, cell cycle microarray data [30]. These microarray experiments were designed to create a comprehensive list of yeast genes whose transcription levels were expressed periodically within the cell cycle. We select one of this profile in which the gene expressions of cell cycle synchronized yeast cultures were collected over 17 time points taken in 10minute intervals. This time series covers more than two complete cycles of cell division. The first time point, related to the M step, is excluded in order to better recover the time relationships present in the G1 step. The true edges of the underlying network were provided by KEGG yeast's cell cycle reference pathway [42].
Green Fluorescent Protein RealTime Gene Expression Data
The time course profiles for a set of 8 genes, part of the SOS pathway of E. coli[31] are selected. Data are produced by a system for realtime monitoring of the transcriptional activity of operons by means of lowcopy reporter plasmids in which a promoter controls GFP (green fluorescent protein). Even if the data contain 50 time points we use only the first 14 points (excluding the first point of the TS data which is zero) avoiding the misguiding flat tails characterizing such gene profiles (the response to the UV stimulus is quick, so very soon mRNAs came back to prestimulus condition). The expression levels are obtained by averaging the replicates.
IRMA network
Two sets of five genes of time course profiles are provided by realtime PCR from an in vivo yeast synthetic network [32]. One set, called Switch ON data set, is the result of the time measurements, every 20 minutes for 5 hours, of the mRNA concentration after shifting cells from glucose to galactose, for a total of 5 profiles of 16 points. The other one, called Switch OFF data set, is the result of the time measurements, every 10 minutes for 3 hours, of the mRNA concentration after shifting cells from galactose to glucose, for a total of 5 profiles of 21 points. The true edges of the underlying network are provided by the experiment design, and are provided as supplementary information from the paper [32].
Algorithms
ARACNE
The ARACNE algorithm has been proposed in [12, 43]. ARACNE is an informationtheoretic algorithm for the reverse engineering of transcriptional networks from steadystate microarray data. ARACNE, just as many other approaches, is based on the assumptions that the expression level of a given gene can be considered as a random variable, and the mutual relationships between them can be derived by statistical dependencies. It defines an edge as an irreducible statistical dependency between gene expression profiles that cannot be explained as an artifact of other statistical dependencies in the network. It is a two steps algorithm: network construction and network pruning. Within the assumption of a twoway network, all statistical dependencies can be inferred from pairwise marginals, and no higher order analysis is needed. ARACNE identifies candidate interactions by estimating pairwise gene expression profile mutual information, I(g_{ i }, g_{ j }) ≡ I_{ ij }, an informationtheoretic measure of relatedness that is zero iff the joint distribution between the expression level of gene i and gene j satisfies P(g_{ i }, g_{ j }) = P(g_{ i })P(g_{ j }). ARACNE estimates MI using a computationally efficient Gaussian Kernel estimator. Since MI is reparameterization invariant, ARACNE copulatransforms (i.e., rankorder) the profiles before MI estimation; the range of these transformed variables is thus between 0 and 1, and their marginal probability distributions are manifestly uniform. This decreases the influence of arbitrary transformations involved in microarray data preprocessing and removes the need to consider positiondependent kernel widths which might be preferable for nonuniformly distributed data. Secondly the MIs are filtered using an appropriate threshold, I_{0} thus removing the most of indirect candidate interactions using a well known information theoretic property, the data processing inequality (DPI). ARACNE eliminate all edges for which the null hypothesis of mutually independent genes cannot be ruled out. To this extent, ARACNE randomly shuffles the expression of genes across the various microarray profiles and evaluate the MI for such manifestly independent genes. The DPI states that if genes g_{1} and g_{3} interact only through a third gene, g_{2}, (i.e., if the interaction network is g_{1} ↔ ... ↔ g_{2} ↔ ... ↔ g_{3} and no alternative path exists between g_{1} and g_{3}), then I(g_{1,}g_{3}) ≤ min(I(g_{1}, g_{2}); I(g_{2}, g_{3})) [44]. Thus the least of the three MIs can come from indirect interactions only, and so it's pruned.
TimeDelayARACNE
TimeDelayARACNE tries to extend to timecourse data ARACNE retrieving time statistical dependency between gene expression profiles. TimeDelayARACNE is a 3 steps algorithm: it first detects, for all genes, the time point of the initial changes in the expression, secondly there is network construction than network pruning.
Step 1
The thresholds are chosen with Open image in new window . In all the reported experiments we used τ_{ up }= 1.2 and consequently τ_{ down }= 0.83. The quantity IcE(g_{ a }) can be used in order to reduce the unuseful influence relations between genes. Indeed, a gene g_{ a }can eventually influence gene g_{ b }only if IcE(g_{ a }) ≤ IcE(g_{ b }) [7].
Step 2
In particular, if the measure Infl(g_{ a }, g_{ b }) is above the the significance threshold, explained below, for a value of κ > 0, then this means that the activation of gene g_{ a }influences the activation of gene g_{ b }at a later time.
In other terms there is a directed link "from" gene g_{ a }"to" gene g_{ b }, this is the way TimeDelayARACNE can recover directed edges. On the contrary, the ARACNE algorithm does not produce directed edges as it corresponds to the case κ = 0, and the Mutual Information is of course symmetric.
We want to show direct gene interactions so under the condition of the perfect choice of experimental time points the best time delay is one because it allows to better capture direct interactions while other delays ideally should evidence more indirect interactions but usually time points are not sharply calibrated to detect such information, so considering few different time points could help in the task. If you consider a too long time delay you can see a correlation between gene a and gene c losing gene b which is regulated by a and regulates c while short time delay can be not sufficient to evidence the connection between gene a and gene b, so using some few delays we try to overcome the above problem. Other approaches based, for example, of conditional mutual information, such as in [48], could of course be exploited.
Step 3
In the last step TimeDelayARACNE uses the DPI twice. In particular the first DPI is useful to split threenodes cliques (triangles) at a single time delay. Whereas the second is applied after the computation of the time delay between each pair of genes as in (8). Just as in the standard ARACNE algorithm, three genes loops are still possible on the basis of a tolerance parameter. In particular triangles are maintained by the algorithm if the difference between the mutual information of the three connections is below the 15% (this the same tolerance parameter adopted in the ARACNE algorithm).
Computational Issues
The computational performance of the TimeDelayARACNE algorithm is influenced by the number of genes, by the mutual information estimation algorithm and by the adopted scheme of bootstrapping for the estimation of the threshold parameter. In particular if the network has n genes and t samples, we have to compute O(Kn^{2}) estimations of the mutual information between two vectors of samples having t elements or less, K being the maximum value of the parameter κ. We adopt a kernelbased estimator of the density of data used in the computation of Mutual Information; it is based on procedure proposed in [58] and implemented in the R package "GenKern" http://cran.rproject.org/. It performs a smoothing of data and an interpolation on a grid of fixed dimensions; the procedure also performs an automatic selection of the kernel bandwidth, by choosing the bandwidth which (approximately) minimizes good quality estimates of the mean integrated squared error [59]. Indeed, there are also other, more recent and elaborate, approaches for estimating entropy and Mutual Information. In particular approaches such as those proposes in [11, 60] deal with entropy estimation in the cases of a small number of highdimensional samples, where the kernelbased density estimator could be rather inefficient.
Therefore, each inner mutual information estimation just depends on t and on the size of the fixed grid, which in all our experiments we fixed at 100 × 100. The algorithms were developed in R and available at the site http://bioinformatics.biogem.it. To have an idea of the computational time required by each network reconstruction, the estimation of the mutual information on a standard platform (Intel Core 2 2, 4 GHz Duo processor with 2 GB RAM) between two expression profiles of size from 10 to 100 points ranges on the average between 0.07 and 0.13 seconds. The whole procedure, apart from the bootstrapping required to estimate the threshold I_{0}, on a network of 50 genes and 50 time points, requires less than 7 minutes. Therefore the most computational demanding step is the bootstrapping, it is needed to compute the threshold I_{0}. It consists in randomly permuting the dataset (the set of expression profiles row values), and then calculating the average mutual information and standard deviation of these random values. Depending on the number of samples in the bootstrap steps, the computational time changes; in all the reported experiments we used a number of 500 bootstrapping samples, this turns out to produce the reconstructed network of 50 genes and 50 time points in about 47 minutes.
Availability and Requirements
The software was implemented in R and can be downloaded at http://bioinformatics.biogem.it or by contacting the corresponding author.
Notes
Acknowledgements
We thank the anonymous reviewers for their help in the improvement of the original manuscript. This work was supported by Italian Ministry MIUR (PRIN 20085CH22F).
Supplementary material
References
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